AdS spacetimes and isometric embeddings

An algebraic global isometric embedding of the nonrotating BTZ black hole is presented. The ambient spacetime is $\mathbb{M}^{2,3}$, the 3+2 dimensional flat spacetime. We also present the analogous embedding for the Euclidean BTZ spacetime and by performing a kind of double analytic continuation construct a 1-parameter family of embeddings of cosmological AdS spacetime into $\mathbb{M}^{2,3}$ which coincide asymptotically with the embedded BTZ manifold of the appropriate mass. Finally we note that the family of embeddings of cosmological AdS$_{n}$ into $\mathbb{M}^{2,n}$ generalises to higher dimensions.